Special least squares solutions of the quaternion matrix equation AX=BAX=B with applications

In this paper, by applying particular structure of the real representations of quaternion matrices and the Moore–Penrose generalized inverse, we derive the expressions of the minimal norm least squares solution, the pure imaginary least squares solution, and the real least squares solution for the quaternion matrix equation AX=BAX=B. The resulting formulas only involve real matrices, which are simpler than those reported in (Yuan et al., 2013). The corresponding algorithms only perform real arithmetic which also consider particular structure of the real representations of quaternion matrices, therefore are very efficient and easily understood. Numerical examples are provided to illustrate the efficiency of our algorithms.